Robustness of Control Barrier Functions for Safety Critical Control

TL;DR

This paper extends control barrier functions to enhance robustness against disturbances, ensuring safety and well-posedness of control systems through new stability conditions and Lipschitz continuity guarantees.

Contribution

It introduces robustness extensions for control barrier functions, providing input-to-state stability conditions and ensuring Lipschitz continuity of the control law.

Findings

Robustness conditions ensure safety under disturbances.

Control laws are Lipschitz continuous, guaranteeing well-posed solutions.

Framework enables safe control synthesis with disturbances present.

Abstract

Barrier functions (also called certificates) have been an important tool for the verification of hybrid systems, and have also played important roles in optimization and multi-objective control. The extension of a barrier function to a controlled system results in a control barrier function. This can be thought of as being analogous to how Sontag extended Lyapunov functions to control Lyapunov functions in order to enable controller synthesis for stabilization tasks. A control barrier function enables controller synthesis for safety requirements specified by forward invariance of a set using a Lyapunov-like condition. This paper develops several important extensions to the notion of a control barrier function. The first involves robustness under perturbations to the vector field defining the system. Input-to-State stability conditions are given that provide for forward invariance, when disturbances are present, of a "relaxation" of set rendered invariant without disturbances. A control barrier function can be combined with a control Lyapunov function in a quadratic program to achieve a control objective subject to safety guarantees. The second result of the paper gives conditions for the control law obtained by solving the quadratic program to be Lipschitz continuous and therefore to gives rise to well-defined solutions of the resulting closed-loop system.